# Properties

 Label 76.3.b.b Level $76$ Weight $3$ Character orbit 76.b Analytic conductor $2.071$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384$$ x^14 - 2*x^13 + x^12 + 14*x^11 - 42*x^10 + 28*x^9 + 132*x^8 - 440*x^7 + 528*x^6 + 448*x^5 - 2688*x^4 + 3584*x^3 + 1024*x^2 - 8192*x + 16384 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( - \beta_{12} + \beta_{6}) q^{6} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{2} - 3) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 6) q^{9}+O(q^{10})$$ q + b3 * q^2 - b7 * q^3 + b5 * q^4 + b12 * q^5 + (-b12 + b6) * q^6 + (-b13 - b8 - b2) * q^7 + (-b12 - b11 - b7 - b2 - 3) * q^8 + (-b13 - b12 + b11 - b10 - b9 + b8 + b7 - b6 + 2*b5 + b2 - 6) * q^9 $$q + \beta_{3} q^{2} - \beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( - \beta_{12} + \beta_{6}) q^{6} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{2} - 3) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 6) q^{9}+ \cdots + ( - 3 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} + \cdots + 3) q^{99}+O(q^{100})$$ q + b3 * q^2 - b7 * q^3 + b5 * q^4 + b12 * q^5 + (-b12 + b6) * q^6 + (-b13 - b8 - b2) * q^7 + (-b12 - b11 - b7 - b2 - 3) * q^8 + (-b13 - b12 + b11 - b10 - b9 + b8 + b7 - b6 + 2*b5 + b2 - 6) * q^9 + (-b10 - b7 - 1) * q^10 + (b13 + b12 + b11 + b9 + b4 - b3 + b2 + 1) * q^11 + (-b13 + b6 + b5 - b4 - 3*b2 + b1 - 1) * q^12 + (b13 - b12 - b8 - 2*b5 + 5) * q^13 + (b13 + b12 - b11 - 2*b8 + b7 + b6 - 2*b5 + b3 + b1 + 3) * q^14 + (b13 + b12 + b11 + b9 - b7 - b4 + 3*b3 + 2*b1 - 1) * q^15 + (b13 - b12 + b10 + b7 - b5 + b4 - 3*b3 - b2 - b1 + 6) * q^16 + (2*b10 + 2*b8 - 2*b6 - 2*b3 + 2*b2 - 2*b1 + 3) * q^17 + (-2*b13 - 2*b11 + b10 - b7 - 4*b3 - 2*b1 + 3) * q^18 - b2 * q^19 + (-b12 - b11 + b9 + b8 + b6 - b4 - b3 - b1 + 2) * q^20 + (2*b13 + 4*b12 + b11 - b9 - b8 + b7 - 2*b6 - 3*b5 + 2*b4 + 5*b3 + 2*b2 - 1) * q^21 + (-b13 - 3*b12 - b11 - b10 + 2*b8 - b6 + 2*b5 - 2*b4 - 2*b3 + 2*b2 + b1) * q^22 + (-b13 - b12 - 2*b11 - b10 + b7 - b6 - b4 - b3 + 3*b2 - 1) * q^23 + (-2*b13 + b12 + b11 - 2*b10 - 3*b9 + b8 + 4*b7 - b6 + b5 + b4 + b3 - b1 - 8) * q^24 + (b13 + b12 - b11 + b10 + b9 - b8 - b7 + b6 - 2*b5 - b2 - 5) * q^25 + (3*b13 + 3*b12 + b11 + 3*b10 + b9 + b8 + b7 - 2*b5 + 4*b3 + 3*b2) * q^26 + (b13 + b12 + 3*b11 + 2*b10 - b9 + 6*b7 + 2*b6 + b4 + 3*b3 + 2*b2 + 1) * q^27 + (5*b13 + 6*b12 + b11 + b10 + 3*b9 - b8 - 3*b7 + b6 - 3*b5 + 2*b4 + 2*b3 - b2 + 2*b1 + 4) * q^28 + (-4*b13 + b11 - 2*b10 - b9 + 3*b8 + b7 + b5 - 2*b4 - b3 - 2*b1 + 1) * q^29 + (-b13 + 3*b11 - b10 - b9 + 3*b8 + 5*b7 - b6 + 6*b5 + 2*b4 - 2*b3 + 7*b2 - 16) * q^30 + (-b13 - b12 - b11 - b9 - b7 - 2*b5 + b4 - 7*b3 - 2*b2 + 1) * q^31 + (b13 + 2*b10 - 2*b9 + 2*b8 - 3*b6 - 3*b5 - b4 + 4*b3 + b2 - b1 + 5) * q^32 + (-b13 - 3*b12 - b11 + 2*b10 + b9 + 2*b8 - b7 + 5*b5 - 3*b3) * q^33 + (-2*b13 - 4*b12 + 2*b11 - 2*b9 - 2*b8 - 2*b6 + 3*b3 + 2*b2 - 6) * q^34 + (-b13 - b12 - b11 - b9 + 2*b7 - 2*b5 - b4 - 3*b3 - 5*b2 + 2*b1 - 1) * q^35 + (-3*b12 - b11 - 2*b10 + b9 - 7*b8 - 2*b7 + 3*b6 - 6*b5 - b4 + 5*b3 - 8*b2 + 3*b1 + 8) * q^36 + (-b13 + b12 - b11 - 2*b10 + b9 - 2*b8 - b7 + 4*b6 + b5 - 2*b4 - 3*b3 - 4*b2 + 2*b1 + 6) * q^37 + (-b9 + b8 + b7 - b6 + b2 - b1) * q^38 + (b13 - b12 - 2*b11 - b10 + 2*b8 - 5*b7 - b6 + 2*b5 + 3*b4 - 5*b3 + 7*b2 - 6*b1 + 3) * q^39 + (-3*b13 - 3*b12 + b11 - 2*b10 + 3*b7 - b6 + 3*b5 - 3*b4 + 2*b3 - 4*b2 + b1 - 14) * q^40 + (3*b13 + 7*b12 + b11 - b9 - 2*b8 + b7 - 2*b6 + 3*b5 - 7*b3 + 2*b2 + 2*b1 + 18) * q^41 + (6*b13 + 5*b12 + 2*b10 + 2*b9 + 2*b8 - 8*b7 - b6 + 2*b4 - 4*b3 + 8*b2 - 2*b1 + 20) * q^42 + (b13 - b12 - 3*b11 - 2*b10 + b9 + 2*b8 - 6*b7 - 2*b6 + 2*b5 - b4 + b3 - 5*b2 - 2*b1 - 1) * q^43 + (-3*b13 - b12 + 2*b11 + b10 + 2*b9 - 2*b8 + 5*b7 + 3*b4 + 3*b3 - b2 - b1 - 6) * q^44 + (-11*b12 - 2*b11 - 2*b10 + 2*b9 - 4*b8 - 2*b7 + 6*b6 - 6*b5 + 8*b3 - 6*b2 + 2*b1 - 12) * q^45 + (2*b13 + 2*b12 + b10 + 6*b9 - 6*b8 - b7 + 2*b6 - 4*b5 + 2*b4 + b3 - 4*b2 + 2*b1 + 7) * q^46 + (-2*b13 - b12 + b10 - 2*b9 - b8 + 7*b7 + b6 - b4 + 3*b3 + 2*b2 - 1) * q^47 + (-b13 + 6*b12 - 4*b11 - 8*b7 - b6 - 3*b5 - b4 - 4*b3 - 3*b2 - 5*b1 + 23) * q^48 + (-4*b12 - 4*b11 + 4*b9 - 4*b8 - 4*b7 + 8*b6 + 6*b5 - 8*b2 + 6*b1 - 18) * q^49 + (2*b13 + 2*b11 - b10 + b7 - 7*b3 + 2*b1 - 3) * q^50 + (-4*b13 - 2*b12 - 4*b11 - 2*b10 - 2*b8 - 7*b7 - 2*b6 + 2*b5 - 6*b4 + 10*b3 - 14*b2 + 2*b1 - 6) * q^51 + (-b13 + 6*b11 - 2*b10 - 2*b9 + 2*b8 - 2*b7 - 3*b6 + 9*b5 + b4 - 4*b3 + 5*b2 + 3*b1 - 23) * q^52 + (4*b13 - 4*b12 - 3*b11 + 4*b10 + 3*b9 - 3*b8 - 3*b7 + 2*b6 - 7*b5 + 2*b4 + 5*b3 - 2*b2 + 5) * q^53 + (-3*b13 + 7*b12 + b11 - b10 - 3*b9 + b8 - 7*b7 - 2*b6 + 6*b5 - 2*b4 - 3*b2 + 4*b1 - 14) * q^54 + (2*b13 - b12 + b10 - 2*b9 + 3*b8 - b7 + b6 + 2*b5 - b4 + 7*b3 + 6*b2 - 2*b1 - 1) * q^55 + (2*b13 - 5*b12 + 3*b11 - 4*b10 - 2*b9 + 10*b8 + b7 - 4*b6 + 8*b5 - 6*b3 + 11*b2 - 3) * q^56 + (-b10 - b8 + b6 - 3*b5 + 3*b3 - b2 + 3) * q^57 + (-10*b13 - b12 - 6*b10 - 2*b9 - 2*b8 + 4*b7 + b6 + 4*b5 - 6*b4 + 8*b3 - 4*b2 - 2*b1 - 12) * q^58 + (3*b13 + b12 + 5*b11 + 4*b10 - 3*b9 + 2*b8 + 4*b7 + 4*b6 - 2*b5 + 5*b4 - 5*b3 + 2*b2 - 2*b1 + 5) * q^59 + (-8*b13 - 4*b12 - 6*b11 - 2*b10 + 2*b9 - 2*b8 - 10*b7 + 4*b5 - 2*b4 - 16*b3 - 8*b2 + 2*b1 + 10) * q^60 + (-8*b13 - 11*b12 - 6*b10 + 2*b8 + 6*b6 + 8*b5 - 4*b4 - 2*b3 - 6*b2 + 2*b1 - 6) * q^61 + (b13 + 2*b12 + b11 + b10 - b9 - b8 + b7 + b6 - 10*b5 + 2*b4 + 2*b3 - b2 - 2*b1 + 32) * q^62 + (4*b13 + 5*b12 + 8*b11 + 3*b10 + 2*b9 - b8 + 7*b7 + 3*b6 + 6*b5 + b4 + 21*b3 + 10*b2 - 2*b1 + 1) * q^63 + (2*b13 + 5*b12 + 7*b11 + 4*b10 - 3*b9 + b8 - 3*b6 + b5 + 5*b4 + 7*b3 + 10*b2 - 5*b1 - 4) * q^64 + (2*b13 + 8*b12 + 4*b11 - 4*b9 + 2*b8 + 4*b7 - 8*b6 - 2*b5 + 4*b4 + 8*b3 + 8*b2 - 2*b1 - 32) * q^65 + (-5*b13 - 10*b12 - b11 - b10 - 3*b9 - 3*b8 + 3*b7 - b6 + 2*b5 + 2*b4 - 7*b2 + 2*b1 - 4) * q^66 + (-4*b13 - 6*b12 - 8*b11 - 2*b10 - 4*b9 + 2*b8 + b7 - 2*b6 + 2*b5 - 2*b4 - 2*b3 - 4*b2 - 6*b1 - 2) * q^67 + (4*b13 + 6*b12 - 2*b11 + 8*b10 + 2*b9 - 6*b8 + 6*b6 - 5*b5 + 2*b4 - 2*b3 + 4*b2 + 2*b1 + 8) * q^68 + (b13 + 5*b12 + 2*b11 + 2*b10 - 2*b9 + 3*b8 + 2*b7 - 6*b6 + 12*b5 + 4*b4 + 6*b2 + 2*b1 + 7) * q^69 + (b13 + 9*b12 + 5*b11 + b10 - 4*b9 + 2*b8 + 8*b7 - 5*b6 - 6*b5 + 6*b4 + 2*b3 + 6*b2 - 5*b1 + 16) * q^70 + (2*b13 + 2*b12 - 2*b10 + 4*b9 - 4*b7 - 2*b6 - 6*b5 + 2*b4 - 18*b3 + 4*b2 + 6*b1 + 2) * q^71 + (11*b13 + 13*b12 + b11 + 6*b10 + 2*b9 + 2*b8 + 13*b7 + 3*b6 - 5*b5 + b4 + 8*b3 + 8*b2 + b1 + 10) * q^72 + (-2*b13 - 2*b12 + 2*b11 - 4*b10 - 2*b9 + 2*b7 + 10*b5 - 2*b3 + 4*b1 + 1) * q^73 + (-b13 + 2*b12 - b11 - 5*b10 + b9 + b8 + 3*b7 + 3*b6 - 2*b5 - 2*b4 + 8*b3 - 7*b2 + 2*b1 - 20) * q^74 + (-b13 - b12 - 3*b11 - 2*b10 + b9 - 4*b7 - 2*b6 - b4 - 3*b3 - 2*b2 - 1) * q^75 + (b12 + b10 - 3*b7 - b6 - b5 + b3 - 2*b1 + 1) * q^76 + (2*b13 + 15*b12 + 6*b10 + 4*b8 - 6*b6 - 20*b5 + 4*b4 + 18*b3 + 6*b2 - 10*b1 + 22) * q^77 + (-18*b12 - 10*b11 + b10 + 8*b9 - 4*b8 - 17*b7 + 8*b6 - 4*b5 - 10*b4 - b3 - 18*b2 + 2*b1 + 17) * q^78 + (3*b13 - 3*b12 - b11 + 2*b10 - 5*b9 + 6*b8 + b7 + 2*b6 - 4*b5 - b4 - 5*b3 + 8*b2 + 2*b1 - 1) * q^79 + (-3*b13 + 4*b12 - b11 + b10 - 3*b9 + b8 + 13*b7 - 5*b6 + 2*b5 - 8*b3 + 7*b2 - 4*b1 - 24) * q^80 + (-2*b13 - 8*b12 - 2*b11 + 2*b10 + 2*b9 + 2*b8 - 2*b7 + 2*b6 - 8*b4 - 24*b3 - 2*b2 - 4*b1 + 25) * q^81 + (9*b13 + 4*b12 - 3*b11 + b10 + 3*b9 + 3*b8 - 15*b7 - b6 - 14*b5 + 6*b4 + 16*b3 + 7*b2 - 2*b1 - 24) * q^82 + (2*b13 + 6*b12 + 4*b11 - 2*b10 + 8*b9 - 4*b8 + 6*b7 - 2*b6 + 2*b5 - 2*b4 + 10*b3 - 6*b2 + 6*b1 - 2) * q^83 + (b13 - 18*b12 - 2*b10 + 4*b9 + 4*b8 - 14*b7 + 5*b6 + 3*b5 - 3*b4 + 10*b3 - b2 + 3*b1 - 21) * q^84 + (4*b13 + 3*b12 - 4*b11 + 6*b10 + 4*b9 - 2*b8 - 4*b7 + 2*b6 + 2*b5 - 4*b4 - 22*b3 - 2*b2 + 8) * q^85 + (b13 - 13*b12 - 3*b11 + b10 - 2*b9 + 8*b8 + 6*b7 - 5*b6 + 2*b5 - 2*b4 - 2*b3 + 4*b2 - 11*b1 - 8) * q^86 + (-3*b13 + 3*b12 + 2*b11 - b10 + 4*b9 - 6*b8 + 3*b7 - b6 - 4*b5 - b4 - 5*b3 + 3*b2 + 8*b1 - 1) * q^87 + (-3*b13 - b12 - 5*b11 - 2*b10 - 2*b9 - 6*b8 + 7*b7 - 3*b6 + 5*b5 - 5*b4 - 8*b3 + 7*b1 + 30) * q^88 + (-5*b13 - 5*b12 - b11 - 4*b10 + b9 - b7 + 6*b6 - 3*b5 - 8*b4 - 13*b3 - 6*b2 - 2*b1 - 2) * q^89 + (2*b13 + 6*b12 + 2*b11 + 7*b10 + 2*b9 + 2*b8 + 19*b7 + 4*b6 + 8*b5 - 4*b4 - 14*b3 - 6*b2 + 4*b1 + 29) * q^90 + (-7*b13 - b12 - b11 - b9 - 6*b8 + 6*b7 - 2*b5 + 7*b4 - 19*b3 - 8*b2 - 6*b1 + 7) * q^91 + (6*b13 - 3*b12 - b11 - 2*b10 + b9 + b8 + 14*b7 - 3*b6 + 3*b5 - 3*b4 - 3*b3 + 10*b2 + 9*b1 + 6) * q^92 + (4*b13 + 18*b12 + 6*b11 - 6*b9 + 2*b8 + 6*b7 - 12*b6 - 6*b5 + 4*b4 + 6*b3 + 12*b2 - 4*b1 - 8) * q^93 + (b13 + 13*b12 + b11 + b10 + 2*b9 - 8*b8 - 6*b7 + b6 - 2*b5 + 2*b4 + 4*b3 - 8*b2 + 5*b1 - 4) * q^94 + (-2*b13 - b12 - 2*b11 - b10 - b8 - b7 - b6 - b4 - b3 - 2*b2 - 1) * q^95 + (4*b13 - 9*b12 - b11 - 6*b10 + b9 - 3*b8 - 6*b7 + 5*b6 - 9*b5 - 5*b4 + 25*b3 - 4*b2 - 3*b1 - 8) * q^96 + (2*b13 + 8*b12 + 2*b11 + 8*b10 - 2*b9 + 8*b8 + 2*b7 - 12*b6 - 12*b5 + 8*b4 + 22*b3 + 12*b2 - 10*b1 + 26) * q^97 + (-8*b12 - 4*b11 - 4*b10 + 8*b7 + 4*b6 + 4*b5 + 4*b4 - 22*b3 - 20*b2 + 8*b1 + 4) * q^98 + (-3*b13 + 3*b12 + 3*b11 + 3*b9 - 6*b8 - 4*b7 - 4*b5 + 3*b4 - 11*b3 - 23*b2 + 4*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9}+O(q^{10})$$ 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9 $$14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9} - 12 q^{10} + 4 q^{12} + 54 q^{13} + 30 q^{14} + 58 q^{16} + 34 q^{17} + 36 q^{18} + 32 q^{20} - 38 q^{21} + 36 q^{22} - 98 q^{24} - 86 q^{25} - 16 q^{26} + 18 q^{28} + 54 q^{29} - 204 q^{30} + 72 q^{32} + 20 q^{33} - 82 q^{34} + 96 q^{36} + 100 q^{37} - 148 q^{40} + 224 q^{41} + 224 q^{42} - 96 q^{44} - 168 q^{45} + 46 q^{46} + 296 q^{48} - 220 q^{49} - 58 q^{50} - 288 q^{52} + 14 q^{53} - 128 q^{54} + 12 q^{56} + 38 q^{57} - 72 q^{58} + 188 q^{60} + 28 q^{61} + 396 q^{62} - 118 q^{64} - 472 q^{65} - 32 q^{66} + 30 q^{68} + 122 q^{69} + 156 q^{70} + 80 q^{72} + 70 q^{73} - 224 q^{74} + 228 q^{77} + 274 q^{78} - 348 q^{80} + 334 q^{81} - 400 q^{82} - 216 q^{84} + 48 q^{85} - 124 q^{86} + 472 q^{88} + 416 q^{90} + 126 q^{92} - 176 q^{93} - 88 q^{94} - 106 q^{96} + 308 q^{97} + 68 q^{98}+O(q^{100})$$ 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9 - 12 * q^10 + 4 * q^12 + 54 * q^13 + 30 * q^14 + 58 * q^16 + 34 * q^17 + 36 * q^18 + 32 * q^20 - 38 * q^21 + 36 * q^22 - 98 * q^24 - 86 * q^25 - 16 * q^26 + 18 * q^28 + 54 * q^29 - 204 * q^30 + 72 * q^32 + 20 * q^33 - 82 * q^34 + 96 * q^36 + 100 * q^37 - 148 * q^40 + 224 * q^41 + 224 * q^42 - 96 * q^44 - 168 * q^45 + 46 * q^46 + 296 * q^48 - 220 * q^49 - 58 * q^50 - 288 * q^52 + 14 * q^53 - 128 * q^54 + 12 * q^56 + 38 * q^57 - 72 * q^58 + 188 * q^60 + 28 * q^61 + 396 * q^62 - 118 * q^64 - 472 * q^65 - 32 * q^66 + 30 * q^68 + 122 * q^69 + 156 * q^70 + 80 * q^72 + 70 * q^73 - 224 * q^74 + 228 * q^77 + 274 * q^78 - 348 * q^80 + 334 * q^81 - 400 * q^82 - 216 * q^84 + 48 * q^85 - 124 * q^86 + 472 * q^88 + 416 * q^90 + 126 * q^92 - 176 * q^93 - 88 * q^94 - 106 * q^96 + 308 * q^97 + 68 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{12} - 2 \nu^{11} + \nu^{10} + 14 \nu^{9} - 42 \nu^{8} + 28 \nu^{7} + 132 \nu^{6} - 440 \nu^{5} + 528 \nu^{4} + 448 \nu^{3} - 640 \nu^{2} + 3584 \nu + 1024 ) / 1024$$ (v^12 - 2*v^11 + v^10 + 14*v^9 - 42*v^8 + 28*v^7 + 132*v^6 - 440*v^5 + 528*v^4 + 448*v^3 - 640*v^2 + 3584*v + 1024) / 1024 $$\beta_{2}$$ $$=$$ $$( - \nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 ) / 20480$$ (-v^13 + 14*v^12 - 9*v^11 + 94*v^10 - 286*v^9 + 844*v^8 - 980*v^7 - 1880*v^6 + 8592*v^5 - 8832*v^4 - 3968*v^3 + 59392*v^2 - 119808*v + 114688) / 20480 $$\beta_{3}$$ $$=$$ $$( - \nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 3584 \nu^{2} - 1024 \nu + 8192 ) / 4096$$ (-v^13 + 2*v^12 - v^11 - 14*v^10 + 42*v^9 - 28*v^8 - 132*v^7 + 440*v^6 - 528*v^5 - 448*v^4 + 2688*v^3 - 3584*v^2 - 1024*v + 8192) / 4096 $$\beta_{4}$$ $$=$$ $$( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} + 4096 ) / 2048$$ (-v^13 + 3*v^11 - 16*v^10 + 14*v^9 + 56*v^8 - 188*v^7 + 176*v^6 + 352*v^5 - 1504*v^4 + 1792*v^3 + 1792*v^2 + 4096) / 2048 $$\beta_{5}$$ $$=$$ $$( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} - 8192 \nu + 6144 ) / 2048$$ (-v^13 + 3*v^11 - 16*v^10 + 14*v^9 + 56*v^8 - 188*v^7 + 176*v^6 + 352*v^5 - 1504*v^4 + 1792*v^3 + 1792*v^2 - 8192*v + 6144) / 2048 $$\beta_{6}$$ $$=$$ $$( - 9 \nu^{13} + 6 \nu^{12} - \nu^{11} + 86 \nu^{10} - 254 \nu^{9} + 476 \nu^{8} - 20 \nu^{7} - 3320 \nu^{6} + 4688 \nu^{5} - 2048 \nu^{4} - 16512 \nu^{3} + 48128 \nu^{2} - 64512 \nu - 22528 ) / 10240$$ (-9*v^13 + 6*v^12 - v^11 + 86*v^10 - 254*v^9 + 476*v^8 - 20*v^7 - 3320*v^6 + 4688*v^5 - 2048*v^4 - 16512*v^3 + 48128*v^2 - 64512*v - 22528) / 10240 $$\beta_{7}$$ $$=$$ $$( 21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032 ) / 20480$$ (21*v^13 + 6*v^12 - 91*v^11 + 246*v^10 - 354*v^9 - 244*v^8 + 2100*v^7 - 1560*v^6 - 112*v^5 + 3392*v^4 - 10112*v^3 + 4608*v^2 - 13312*v + 172032) / 20480 $$\beta_{8}$$ $$=$$ $$( 9 \nu^{13} - 166 \nu^{12} - 159 \nu^{11} + 1354 \nu^{10} - 2146 \nu^{9} - 1116 \nu^{8} + 14740 \nu^{7} - 31240 \nu^{6} + 2992 \nu^{5} + 98048 \nu^{4} - 217728 \nu^{3} + \cdots - 1052672 ) / 20480$$ (9*v^13 - 166*v^12 - 159*v^11 + 1354*v^10 - 2146*v^9 - 1116*v^8 + 14740*v^7 - 31240*v^6 + 2992*v^5 + 98048*v^4 - 217728*v^3 + 161792*v^2 + 402432*v - 1052672) / 20480 $$\beta_{9}$$ $$=$$ $$( 19 \nu^{13} - 126 \nu^{12} - 189 \nu^{11} + 1074 \nu^{10} - 1006 \nu^{9} - 2556 \nu^{8} + 15020 \nu^{7} - 22280 \nu^{6} - 11408 \nu^{5} + 107968 \nu^{4} - 167808 \nu^{3} + \cdots - 991232 ) / 20480$$ (19*v^13 - 126*v^12 - 189*v^11 + 1074*v^10 - 1006*v^9 - 2556*v^8 + 15020*v^7 - 22280*v^6 - 11408*v^5 + 107968*v^4 - 167808*v^3 + 26112*v^2 + 535552*v - 991232) / 20480 $$\beta_{10}$$ $$=$$ $$( 39 \nu^{13} + 114 \nu^{12} - 569 \nu^{11} + 674 \nu^{10} + 954 \nu^{9} - 7036 \nu^{8} + 12380 \nu^{7} + 1400 \nu^{6} - 49168 \nu^{5} + 99648 \nu^{4} - 57728 \nu^{3} - 163328 \nu^{2} + \cdots - 69632 ) / 20480$$ (39*v^13 + 114*v^12 - 569*v^11 + 674*v^10 + 954*v^9 - 7036*v^8 + 12380*v^7 + 1400*v^6 - 49168*v^5 + 99648*v^4 - 57728*v^3 - 163328*v^2 + 453632*v - 69632) / 20480 $$\beta_{11}$$ $$=$$ $$( 11 \nu^{13} - 38 \nu^{12} + 27 \nu^{11} + 106 \nu^{10} - 318 \nu^{9} + 180 \nu^{8} + 1036 \nu^{7} - 3176 \nu^{6} + 2288 \nu^{5} + 4288 \nu^{4} - 11904 \nu^{3} + 2560 \nu^{2} + 46080 \nu - 77824 ) / 4096$$ (11*v^13 - 38*v^12 + 27*v^11 + 106*v^10 - 318*v^9 + 180*v^8 + 1036*v^7 - 3176*v^6 + 2288*v^5 + 4288*v^4 - 11904*v^3 + 2560*v^2 + 46080*v - 77824) / 4096 $$\beta_{12}$$ $$=$$ $$( - 3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 ) / 1024$$ (-3*v^13 + 9*v^12 - v^11 - 39*v^10 + 112*v^9 - 82*v^8 - 328*v^7 + 1012*v^6 - 952*v^5 - 1168*v^4 + 4672*v^3 - 4736*v^2 - 7680*v + 12288) / 1024 $$\beta_{13}$$ $$=$$ $$( - 43 \nu^{13} - 18 \nu^{12} + 293 \nu^{11} - 578 \nu^{10} - 98 \nu^{9} + 3292 \nu^{8} - 8140 \nu^{7} + 3400 \nu^{6} + 22416 \nu^{5} - 56256 \nu^{4} + 50816 \nu^{3} + 78336 \nu^{2} + \cdots + 57344 ) / 10240$$ (-43*v^13 - 18*v^12 + 293*v^11 - 578*v^10 - 98*v^9 + 3292*v^8 - 8140*v^7 + 3400*v^6 + 22416*v^5 - 56256*v^4 + 50816*v^3 + 78336*v^2 - 308224*v + 57344) / 10240
 $$\nu$$ $$=$$ $$( -\beta_{5} + \beta_{4} + 1 ) / 4$$ (-b5 + b4 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 2\beta_{3} + \beta_1 ) / 2$$ (b5 - 2*b3 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2$$ (b13 + b11 + b10 + b9 - b8 + b7 + b6 + b4 + 2*b3 + b2 + b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta _1 + 8 ) / 2$$ (b13 - 2*b12 - b11 + b10 + b9 - b8 - 3*b7 - b6 - b5 - 2*b4 + 2*b3 + 3*b2 + b1 + 8) / 2 $$\nu^{5}$$ $$=$$ $$( 3 \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 7 \beta _1 + 18 ) / 2$$ (3*b13 + 4*b12 + 3*b11 + 3*b10 + b9 + b8 + 3*b7 - 7*b6 - 5*b5 + 6*b4 - 4*b3 + 11*b2 - 7*b1 + 18) / 2 $$\nu^{6}$$ $$=$$ $$( \beta_{13} + 2 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 7 \beta_{8} + 9 \beta_{7} - 17 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + \beta _1 - 20 ) / 2$$ (b13 + 2*b12 - 5*b11 - 3*b10 - 3*b9 + 7*b8 + 9*b7 - 17*b6 - 5*b5 - 2*b4 - 6*b3 + 3*b2 + b1 - 20) / 2 $$\nu^{7}$$ $$=$$ $$( 7 \beta_{13} + 4 \beta_{12} - \beta_{11} - \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} - 40 \beta_{3} - 25 \beta_{2} + 9 \beta _1 + 2 ) / 2$$ (7*b13 + 4*b12 - b11 - b10 + 17*b9 - 11*b8 + 39*b7 + 9*b6 + 15*b5 + 2*b4 - 40*b3 - 25*b2 + 9*b1 + 2) / 2 $$\nu^{8}$$ $$=$$ $$( - 23 \beta_{13} - 54 \beta_{12} - 53 \beta_{11} - 35 \beta_{10} + 29 \beta_{9} - 33 \beta_{8} - 23 \beta_{7} + 39 \beta_{6} - \beta_{5} - 34 \beta_{4} + 82 \beta_{3} - 53 \beta_{2} + 29 \beta _1 - 188 ) / 2$$ (-23*b13 - 54*b12 - 53*b11 - 35*b10 + 29*b9 - 33*b8 - 23*b7 + 39*b6 - b5 - 34*b4 + 82*b3 - 53*b2 + 29*b1 - 188) / 2 $$\nu^{9}$$ $$=$$ $$( - 69 \beta_{13} + 52 \beta_{12} + 19 \beta_{11} - 61 \beta_{10} + 29 \beta_{9} - 15 \beta_{8} - 37 \beta_{7} + 5 \beta_{6} + 159 \beta_{5} - 34 \beta_{4} - 56 \beta_{3} + 59 \beta_{2} - 11 \beta _1 + 254 ) / 2$$ (-69*b13 + 52*b12 + 19*b11 - 61*b10 + 29*b9 - 15*b8 - 37*b7 + 5*b6 + 159*b5 - 34*b4 - 56*b3 + 59*b2 - 11*b1 + 254) / 2 $$\nu^{10}$$ $$=$$ $$( - 75 \beta_{13} + 210 \beta_{12} + 127 \beta_{11} - 103 \beta_{10} - 55 \beta_{9} + 115 \beta_{8} + 149 \beta_{7} - 101 \beta_{6} - 133 \beta_{5} + 38 \beta_{4} + 490 \beta_{3} + 255 \beta_{2} - 31 \beta _1 - 316 ) / 2$$ (-75*b13 + 210*b12 + 127*b11 - 103*b10 - 55*b9 + 115*b8 + 149*b7 - 101*b6 - 133*b5 + 38*b4 + 490*b3 + 255*b2 - 31*b1 - 316) / 2 $$\nu^{11}$$ $$=$$ $$( 23 \beta_{13} + 340 \beta_{12} + 223 \beta_{11} - 97 \beta_{10} + 33 \beta_{9} + 5 \beta_{8} + 311 \beta_{7} + 57 \beta_{6} - 189 \beta_{5} - 130 \beta_{4} - 888 \beta_{3} - 89 \beta_{2} + 281 \beta _1 + 1886 ) / 2$$ (23*b13 + 340*b12 + 223*b11 - 97*b10 + 33*b9 + 5*b8 + 311*b7 + 57*b6 - 189*b5 - 130*b4 - 888*b3 - 89*b2 + 281*b1 + 1886) / 2 $$\nu^{12}$$ $$=$$ $$( 137 \beta_{13} - 86 \beta_{12} - 85 \beta_{11} + 61 \beta_{10} + 317 \beta_{9} - 481 \beta_{8} + 201 \beta_{7} + 775 \beta_{6} - 1513 \beta_{5} + 414 \beta_{4} - 1118 \beta_{3} - 373 \beta_{2} + 213 \beta _1 - 2684 ) / 2$$ (137*b13 - 86*b12 - 85*b11 + 61*b10 + 317*b9 - 481*b8 + 201*b7 + 775*b6 - 1513*b5 + 414*b4 - 1118*b3 - 373*b2 + 213*b1 - 2684) / 2 $$\nu^{13}$$ $$=$$ $$( - 781 \beta_{13} - 620 \beta_{12} - 405 \beta_{11} - 453 \beta_{10} - 75 \beta_{9} - 519 \beta_{8} - 1645 \beta_{7} + 189 \beta_{6} + 1567 \beta_{5} - 1290 \beta_{4} - 4648 \beta_{3} - 109 \beta_{2} + 909 \beta _1 - 6618 ) / 2$$ (-781*b13 - 620*b12 - 405*b11 - 453*b10 - 75*b9 - 519*b8 - 1645*b7 + 189*b6 + 1567*b5 - 1290*b4 - 4648*b3 - 109*b2 + 909*b1 - 6618) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 −1.92254 + 0.551226i −1.92254 − 0.551226i −1.89728 + 0.632718i −1.89728 − 0.632718i −0.0607713 + 1.99908i −0.0607713 − 1.99908i 0.645572 + 1.89294i 0.645572 − 1.89294i 0.711746 + 1.86907i 0.711746 − 1.86907i 1.57398 + 1.23393i 1.57398 − 1.23393i 1.94929 + 0.447510i 1.94929 − 0.447510i
−1.92254 0.551226i 0.644704i 3.39230 + 2.11950i −2.32715 0.355377 1.23947i 8.62924i −5.35350 5.94475i 8.58436 4.47404 + 1.28279i
39.2 −1.92254 + 0.551226i 0.644704i 3.39230 2.11950i −2.32715 0.355377 + 1.23947i 8.62924i −5.35350 + 5.94475i 8.58436 4.47404 1.28279i
39.3 −1.89728 0.632718i 5.34370i 3.19934 + 2.40089i 5.79268 −3.38106 + 10.1385i 5.87536i −4.55095 6.57943i −19.5551 −10.9903 3.66514i
39.4 −1.89728 + 0.632718i 5.34370i 3.19934 2.40089i 5.79268 −3.38106 10.1385i 5.87536i −4.55095 + 6.57943i −19.5551 −10.9903 + 3.66514i
39.5 −0.0607713 1.99908i 5.37609i −3.99261 + 0.242973i −5.82257 10.7472 0.326712i 5.45132i 0.728358 + 7.96677i −19.9023 0.353845 + 11.6398i
39.6 −0.0607713 + 1.99908i 5.37609i −3.99261 0.242973i −5.82257 10.7472 + 0.326712i 5.45132i 0.728358 7.96677i −19.9023 0.353845 11.6398i
39.7 0.645572 1.89294i 0.820457i −3.16647 2.44406i −2.38184 −1.55308 0.529664i 12.3764i −6.67066 + 4.41614i 8.32685 −1.53765 + 4.50868i
39.8 0.645572 + 1.89294i 0.820457i −3.16647 + 2.44406i −2.38184 −1.55308 + 0.529664i 12.3764i −6.67066 4.41614i 8.32685 −1.53765 4.50868i
39.9 0.711746 1.86907i 4.44946i −2.98683 2.66060i 4.97973 −8.31634 3.16688i 12.2628i −7.09872 + 3.68892i −10.7977 3.54430 9.30745i
39.10 0.711746 + 1.86907i 4.44946i −2.98683 + 2.66060i 4.97973 −8.31634 + 3.16688i 12.2628i −7.09872 3.68892i −10.7977 3.54430 + 9.30745i
39.11 1.57398 1.23393i 2.90118i 0.954817 3.88437i 3.66290 3.57986 + 4.56639i 1.93414i −3.29019 7.29209i 0.583162 5.76533 4.51977i
39.12 1.57398 + 1.23393i 2.90118i 0.954817 + 3.88437i 3.66290 3.57986 4.56639i 1.93414i −3.29019 + 7.29209i 0.583162 5.76533 + 4.51977i
39.13 1.94929 0.447510i 3.19988i 3.59947 1.74465i −3.90374 −1.43198 6.23749i 2.64664i 6.23566 5.01164i −1.23921 −7.60953 + 1.74696i
39.14 1.94929 + 0.447510i 3.19988i 3.59947 + 1.74465i −3.90374 −1.43198 + 6.23749i 2.64664i 6.23566 + 5.01164i −1.23921 −7.60953 1.74696i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.b.b 14
3.b odd 2 1 684.3.g.b 14
4.b odd 2 1 inner 76.3.b.b 14
8.b even 2 1 1216.3.d.d 14
8.d odd 2 1 1216.3.d.d 14
12.b even 2 1 684.3.g.b 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 1.a even 1 1 trivial
76.3.b.b 14 4.b odd 2 1 inner
684.3.g.b 14 3.b odd 2 1
684.3.g.b 14 12.b even 2 1
1216.3.d.d 14 8.b even 2 1
1216.3.d.d 14 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} + 97T_{3}^{12} + 3595T_{3}^{10} + 63443T_{3}^{8} + 539872T_{3}^{6} + 1940896T_{3}^{4} + 1665792T_{3}^{2} + 393984$$ acting on $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} - 2 T^{13} + T^{12} + 14 T^{11} + \cdots + 16384$$
$3$ $$T^{14} + 97 T^{12} + 3595 T^{10} + \cdots + 393984$$
$5$ $$(T^{7} - 66 T^{5} - 28 T^{4} + 1337 T^{3} + \cdots - 13312)^{2}$$
$7$ $$T^{14} + 453 T^{12} + \cdots + 46106276299$$
$11$ $$T^{14} + 880 T^{12} + \cdots + 94488337600$$
$13$ $$(T^{7} - 27 T^{6} - 293 T^{5} + \cdots - 1265920)^{2}$$
$17$ $$(T^{7} - 17 T^{6} - 1107 T^{5} + \cdots - 69101055)^{2}$$
$19$ $$(T^{2} + 19)^{7}$$
$23$ $$T^{14} + \cdots + 683970816311296$$
$29$ $$(T^{7} - 27 T^{6} - 2901 T^{5} + \cdots - 112127472)^{2}$$
$31$ $$T^{14} + 4048 T^{12} + \cdots + 11\!\cdots\!00$$
$37$ $$(T^{7} - 50 T^{6} - 1508 T^{5} + \cdots + 914894720)^{2}$$
$41$ $$(T^{7} - 112 T^{6} + \cdots + 18882293760)^{2}$$
$43$ $$T^{14} + 13244 T^{12} + \cdots + 58\!\cdots\!24$$
$47$ $$T^{14} + 8204 T^{12} + \cdots + 64\!\cdots\!04$$
$53$ $$(T^{7} - 7 T^{6} - 6645 T^{5} + \cdots + 18228603200)^{2}$$
$59$ $$T^{14} + 30513 T^{12} + \cdots + 50\!\cdots\!00$$
$61$ $$(T^{7} - 14 T^{6} + \cdots + 522558358600)^{2}$$
$67$ $$T^{14} + 27889 T^{12} + \cdots + 29\!\cdots\!76$$
$71$ $$T^{14} + 37812 T^{12} + \cdots + 49\!\cdots\!00$$
$73$ $$(T^{7} - 35 T^{6} - 5907 T^{5} + \cdots - 77971926925)^{2}$$
$79$ $$T^{14} + 38740 T^{12} + \cdots + 18\!\cdots\!00$$
$83$ $$T^{14} + 62788 T^{12} + \cdots + 34\!\cdots\!76$$
$89$ $$(T^{7} - 23640 T^{5} + \cdots - 88310345728)^{2}$$
$97$ $$(T^{7} - 154 T^{6} + \cdots + 4410892984320)^{2}$$